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id	title	challengeType	forumTopicId	dashedName
5900f4ff1000cf542c510011	Problem 402: Integer-valued polynomials	5	302070	problem-402-integer-valued-polynomials

--description--

It can be shown that the polynomial n^4 + 4n^3 + 2n^2 + 5n is a multiple of 6 for every integer n. It can also be shown that 6 is the largest integer satisfying this property.

Define M(a, b, c) as the maximum m such that n^4 + an^3 + bn^2 + cn is a multiple of m for all integers n. For example, M(4, 2, 5) = 6.

Also, define S(N) as the sum of M(a, b, c) for all 0 < a, b, c ≤ N.

We can verify that S(10) = 1\\,972 and S(10\\,000) = 2\\,024\\,258\\,331\\,114.

Let F_k be the Fibonacci sequence:

F_0 = 0, F_1 = 1 and
F_k = F_{k - 1} + F_{k - 2} for k ≥ 2.

Find the last 9 digits of \sum S(F_k) for 2 ≤ k ≤ 1\\,234\\,567\\,890\\,123.

--hints--

integerValuedPolynomials() should return 356019862.

assert.strictEqual(integerValuedPolynomials(), 356019862);

--seed--

--seed-contents--

function integerValuedPolynomials() {

  return true;
}

integerValuedPolynomials();

--solutions--

// solution required

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